Problem 30 - General Chemistry: Principles and Modern Applications

Question

A 72.8 L constant-volume cylinder containing \(7.41 \mathrm{g}\) He is heated until the pressure reaches 3.50 atm. What is the final temperature in degrees Celsius?

Step-by-Step Solution

Answer

The final temperature of the gas is \(\approx 1977.17\) degrees Celsius.

Step 1: Convert mass of Helium to moles

To apply the Ideal Gas Law, the number of moles of the gas is needed. Helium (He) has a molar mass of \(4.0026 \mathrm{g / mol}\) approximately, so use the given mass \(7.41 \mathrm{g}\) to calculate the number of moles \(n\) using the formula: \(n = \frac{mass}{molar~mass}\) .

Step 2: Convert pressure to the standard unit

Convert the given pressure from atmospheres to Pascal which is the standard unit of pressure for the Ideal Gas Law. The conversion factor is \(1 \text{ atm} = 1.01325 \times 10^5 \text{ Pa}\). So multiply the given pressure by the conversion factor.

Step 3: Apply the Ideal Gas Law

Substitute the values of pressure \(P\), volume \(V\), moles \(n\) and the gas constant \(R = 8.314 \text{ J/mol K}\) into the Ideal Gas law and solve for the final temperature \(T\). Remember the temperature will be in Kelvin.

Step 4: Convert final temperature to Celsius

Convert the final temperature in Kelvin to degrees Celsius using the formula: \(T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15\)

Key Concepts

Moles Calculation

The Ideal Gas Law is a crucial part of understanding gas behaviors, and it often requires calculating the number of moles. This can be done using the mass of the gas and its molar mass. In this regard, Helium is quite straightforward as it has a molar mass of approximately 4.0026 g/mol.
If you are given a mass of helium gas, say 7.41 g, you can find the number of moles using the formula: \( n = \frac{\text{mass}}{\text{molar mass}} \).
Plug in the values: \( n = \frac{7.41 \text{ g}}{4.0026 \text{ g/mol}} \)
Doing this math will give you the number of moles of helium in the cylinder, which is a critical component for further calculations using the Ideal Gas Law.

Pressure Conversion

Pressure is another essential factor in the Ideal Gas Law. Atmospheric pressure units such as atmospheres (atm) are not the standard SI units used in calculations. The standard unit for pressure in the Ideal Gas Law is Pascal (Pa). Therefore, a conversion is necessary.
The conversion factor between these units is \(1 \text{ atm} = 1.01325 \times 10^5 \text{ Pa} \).
This means that if a problem gives you pressure in atm, you multiply it by this conversion factor to convert it to Pa.

• This conversion ensures you are using the correct units for all terms in the Ideal Gas Law equation.
• Given a pressure of 3.50 atm, convert it to Pa by multiplying: \(3.50 \times 1.01325 \times 10^5 \text{ Pa/atm} \).

This step is vital for accurate application of the Ideal Gas Law.

Temperature Conversion

Temperature in the Ideal Gas Law must be in Kelvin to ensure that calculations are accurate. While some problems might present temperature in Celsius, the conversion to Kelvin is a straightforward addition.
The conversion formula is: \( T_{\text{Kelvin}} = T_{\text{Celsius}} + 273.15 \).
The Kelvin scale starts at absolute zero, making it suitable for calculations in physics, especially gases.

• After solving for the temperature in Kelvin using the Ideal Gas Law, you might need to convert it back to Celsius, particularly if the question specifies Celsius.
• Convert Kelvin to Celsius using: \( T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15 \).

This conversion helps to bring back the temperature readings into a more familiar form when reporting final results.